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# Absolute Value

Absolute value is a mathematician's way of judging numbers by their magnitude rather than their positive/negative value. It is the distance of the number from 0 on a number line.

# Absolute Value Problem Solving

Let $$x=70$$ be one root of the equation $$2\lvert x-10\rvert=x+a,$$ where $$a$$ is a constant. Then what is the other root?

Let $$S$$ be the solution set of the equation $$\lvert 2x-1 \rvert + 3\lvert x-29\rvert=5x-88.$$ Then what is $$S?$$

How many ordered pairs $$(x, y)$$ are there such that $$x$$ is a nonzero integer that satisfies $$-8\leq x \leq 8$$, $$y$$ is a nonzero integer that satisfies $$-2\leq y \leq 2$$, and $\lvert x+y \rvert = \lvert x \rvert + \lvert y \rvert?$

Details and assumptions

For an ordered pair of integers $$(a,b)$$, the order of the integers matter. The ordered pair $$(1, 2)$$ is different from the ordered pair $$(2,1)$$.

How many ordered triples of integers $$(a, b, c)$$ subject to $$-20 < a < b < c < 20,$$ are there, such that

$\left| a + b + c \right | = |a| + |b| + |c|?$

If $$20 < a \leq 21$$, what is the value of $\bigg| 2-a- \big| 6- \lvert a-20 \rvert \big| \bigg| ?$

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